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Theorem nmoofval 30282
Description: The operator norm function. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoofval.1 𝑋 = (BaseSetβ€˜π‘ˆ)
nmoofval.2 π‘Œ = (BaseSetβ€˜π‘Š)
nmoofval.3 𝐿 = (normCVβ€˜π‘ˆ)
nmoofval.4 𝑀 = (normCVβ€˜π‘Š)
nmoofval.6 𝑁 = (π‘ˆ normOpOLD π‘Š)
Assertion
Ref Expression
nmoofval ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ 𝑁 = (𝑑 ∈ (π‘Œ ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )))
Distinct variable groups:   π‘₯,𝑑,𝑧,π‘ˆ   𝑑,π‘Š,π‘₯,𝑧   𝑑,𝑋,𝑧   𝑑,π‘Œ,π‘₯   𝑑,𝐿   𝑑,𝑀
Allowed substitution hints:   𝐿(π‘₯,𝑧)   𝑀(π‘₯,𝑧)   𝑁(π‘₯,𝑧,𝑑)   𝑋(π‘₯)   π‘Œ(𝑧)

Proof of Theorem nmoofval
Dummy variables 𝑒 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmoofval.6 . 2 𝑁 = (π‘ˆ normOpOLD π‘Š)
2 fveq2 6890 . . . . . 6 (𝑒 = π‘ˆ β†’ (BaseSetβ€˜π‘’) = (BaseSetβ€˜π‘ˆ))
3 nmoofval.1 . . . . . 6 𝑋 = (BaseSetβ€˜π‘ˆ)
42, 3eqtr4di 2788 . . . . 5 (𝑒 = π‘ˆ β†’ (BaseSetβ€˜π‘’) = 𝑋)
54oveq2d 7427 . . . 4 (𝑒 = π‘ˆ β†’ ((BaseSetβ€˜π‘€) ↑m (BaseSetβ€˜π‘’)) = ((BaseSetβ€˜π‘€) ↑m 𝑋))
6 fveq2 6890 . . . . . . . . . . 11 (𝑒 = π‘ˆ β†’ (normCVβ€˜π‘’) = (normCVβ€˜π‘ˆ))
7 nmoofval.3 . . . . . . . . . . 11 𝐿 = (normCVβ€˜π‘ˆ)
86, 7eqtr4di 2788 . . . . . . . . . 10 (𝑒 = π‘ˆ β†’ (normCVβ€˜π‘’) = 𝐿)
98fveq1d 6892 . . . . . . . . 9 (𝑒 = π‘ˆ β†’ ((normCVβ€˜π‘’)β€˜π‘§) = (πΏβ€˜π‘§))
109breq1d 5157 . . . . . . . 8 (𝑒 = π‘ˆ β†’ (((normCVβ€˜π‘’)β€˜π‘§) ≀ 1 ↔ (πΏβ€˜π‘§) ≀ 1))
1110anbi1d 628 . . . . . . 7 (𝑒 = π‘ˆ β†’ ((((normCVβ€˜π‘’)β€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§))) ↔ ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))))
124, 11rexeqbidv 3341 . . . . . 6 (𝑒 = π‘ˆ β†’ (βˆƒπ‘§ ∈ (BaseSetβ€˜π‘’)(((normCVβ€˜π‘’)β€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§))) ↔ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))))
1312abbidv 2799 . . . . 5 (𝑒 = π‘ˆ β†’ {π‘₯ ∣ βˆƒπ‘§ ∈ (BaseSetβ€˜π‘’)(((normCVβ€˜π‘’)β€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))} = {π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))})
1413supeq1d 9443 . . . 4 (𝑒 = π‘ˆ β†’ sup({π‘₯ ∣ βˆƒπ‘§ ∈ (BaseSetβ€˜π‘’)(((normCVβ€˜π‘’)β€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))}, ℝ*, < ) = sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))}, ℝ*, < ))
155, 14mpteq12dv 5238 . . 3 (𝑒 = π‘ˆ β†’ (𝑑 ∈ ((BaseSetβ€˜π‘€) ↑m (BaseSetβ€˜π‘’)) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ (BaseSetβ€˜π‘’)(((normCVβ€˜π‘’)β€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )) = (𝑑 ∈ ((BaseSetβ€˜π‘€) ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )))
16 fveq2 6890 . . . . . 6 (𝑀 = π‘Š β†’ (BaseSetβ€˜π‘€) = (BaseSetβ€˜π‘Š))
17 nmoofval.2 . . . . . 6 π‘Œ = (BaseSetβ€˜π‘Š)
1816, 17eqtr4di 2788 . . . . 5 (𝑀 = π‘Š β†’ (BaseSetβ€˜π‘€) = π‘Œ)
1918oveq1d 7426 . . . 4 (𝑀 = π‘Š β†’ ((BaseSetβ€˜π‘€) ↑m 𝑋) = (π‘Œ ↑m 𝑋))
20 fveq2 6890 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (normCVβ€˜π‘€) = (normCVβ€˜π‘Š))
21 nmoofval.4 . . . . . . . . . . 11 𝑀 = (normCVβ€˜π‘Š)
2220, 21eqtr4di 2788 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (normCVβ€˜π‘€) = 𝑀)
2322fveq1d 6892 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)) = (π‘€β€˜(π‘‘β€˜π‘§)))
2423eqeq2d 2741 . . . . . . . 8 (𝑀 = π‘Š β†’ (π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)) ↔ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§))))
2524anbi2d 627 . . . . . . 7 (𝑀 = π‘Š β†’ (((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§))) ↔ ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))))
2625rexbidv 3176 . . . . . 6 (𝑀 = π‘Š β†’ (βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§))) ↔ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))))
2726abbidv 2799 . . . . 5 (𝑀 = π‘Š β†’ {π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))} = {π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))})
2827supeq1d 9443 . . . 4 (𝑀 = π‘Š β†’ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))}, ℝ*, < ) = sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < ))
2919, 28mpteq12dv 5238 . . 3 (𝑀 = π‘Š β†’ (𝑑 ∈ ((BaseSetβ€˜π‘€) ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )) = (𝑑 ∈ (π‘Œ ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )))
30 df-nmoo 30265 . . 3 normOpOLD = (𝑒 ∈ NrmCVec, 𝑀 ∈ NrmCVec ↦ (𝑑 ∈ ((BaseSetβ€˜π‘€) ↑m (BaseSetβ€˜π‘’)) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ (BaseSetβ€˜π‘’)(((normCVβ€˜π‘’)β€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )))
31 ovex 7444 . . . 4 (π‘Œ ↑m 𝑋) ∈ V
3231mptex 7226 . . 3 (𝑑 ∈ (π‘Œ ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )) ∈ V
3315, 29, 30, 32ovmpo 7570 . 2 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ (π‘ˆ normOpOLD π‘Š) = (𝑑 ∈ (π‘Œ ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )))
341, 33eqtrid 2782 1 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ 𝑁 = (𝑑 ∈ (π‘Œ ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {cab 2707  βˆƒwrex 3068   class class class wbr 5147   ↦ cmpt 5230  β€˜cfv 6542  (class class class)co 7411   ↑m cmap 8822  supcsup 9437  1c1 11113  β„*cxr 11251   < clt 11252   ≀ cle 11253  NrmCVeccnv 30104  BaseSetcba 30106  normCVcnmcv 30110   normOpOLD cnmoo 30261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-sup 9439  df-nmoo 30265
This theorem is referenced by:  nmooval  30283  hhnmoi  31421
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